Study on Fuzzy Cognitive Map Structures through
Category Theory
G.Jenitha1, Dr. Shenbaga Ezhil2, Dr. A. Kumaravel3
1
Department of Mathematics, AMET University, Kanathur, Chennai.
2
Department of Mathematics, Jeppiaar Institute of Technology, Chennai.
3
Dean, School of computing, Bharath University, Chennai.
Abstract: In this article we have introduced Fuzzy Cognitive Map (FCM) as a Category model also we have introduced category concepts in Fuzzy Cognitive Mapping .Coined a new Formulation in FCM Category Advantages and limitations are discussed.
I. INTRODUCTION
A. Fuzzy Cognitive Map (FCM)
In 1965, L.A. Zadeh has introduced a mathematical model called Fuzzy Cognitive Maps. After a decade in the year 1976, Political scientist R. Axelord used this fuzzy model to study decision making in social and political systems .Then B. Kosko enhanced the power of cognitive maps considering fuzzy values for the concepts of the cognitive map and fuzzy degrees of interrelationships between concepts. FCMs can successfully represent knowledge and human experience, introduced concepts to represent the essential elements and the cause and effect relationships among the concepts to model the behavior of any system. It is a very convenient simple and powerful tool, which is used in numerous fields such as social, economical and medical etc. illustrated by W.B. Vasantha Kandasamy in her book, “Application of Fuzzy Models in Social Sciences”. In this paper we recall the notion of Fuzzy Cognitive Maps (FCMs), which was introduced by Bart Kosko in the year 1986.
A fuzzy cognitive map (FCM) can be understood as a graphical representation of the knowledge about or the perception of a given system. Fuzzy cognitive mapping is a combination of fuzzy logic and cognitive mapping. Cognitive mapping is based on graph theory, which is also the basis of most calculations and indices. A FCM consists of factors (concepts / nodes) which represent the important elements of the mapped system. The directed lines labeled with fuzzy values show the strength of the causal conditions between the factors. A fuzzy cognitive map is a model of system structure.
B. Category Theory
Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations in 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure. Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed ges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, and the arrows need not be functions. Any way of form aliasing a mathematical concept such that it meets the basic conditions on the behavior of objects and arrows is a valid category and all the results of category theory apply to it.
C. Applications of FCM
Fuzzy Cognitive Maps (FCMs) have been applied in several scientific areas such as:
1) Medicine
2) Engineering
3) Business and Management
4) Production Systems
D. Applications of Category Theory
The above is nice and all, but it’s still just sort of hey-take-my-word-for-it, which is not so satisfying. Here are some actual examples:
1) Category theory has been used to study grammar and human language.
2) In building a spreadsheet application.
3) As a descriptive tool in neuroscience.
4) In the analysis and design of cognitive neural network architectures.
5) Many applications in graduate level mathematics (i.e. stuff I don’t understand) and physics.
6) In programming languages, especially Haskell and most famously monads, but also, for instance, a typed assembly language and work on the typed lambda calculus.
7) Generating program optimizations.
8) To model systems of interacting agents.
9) To generalize sorting algorithms.
10) To understand collaborative text editing.
11) To understand optimal play in sequential games like chess.
12) To formalize the notion of algorithm.
13) In the study of analogy.
14) As “a language for experimental design patterns” and “a new vocabulary in which to think and communicate.”
15) In definitions of emergence and discussions of biology.
16) Generally speaking, there seems to be a cabal of radical category theorists, led by John Baez, who are reinterpreting anything interesting in category theoretic terms. (The trend reminds me of reinventing the wheel for the nth time in the nth newest programming language.)
II. FORMULATION OF FCM CATEGORY (FC)
A. Objects
1) A collection of attributes /concepts /notes (ie) Att (FC)
B. Morphism
1) A collection of Arrows /Relations (A) (ie) Rel (FC)
2) Each Arrows A has a source ‘a’ and target ‘b’ attribute (ie) A: a
bA1
C. Compositions
1) If a, b and c are any three attributes then Rel (a,b)
Rel (b,c)
Rel (a,c)this operation is written as composition of Arrows
a b c
D. Identity Morphism (
X)1) every attributes (relation )X, there exist a morphism
: X
X is called the identity morphism for X such that for everymorphism A : a
b, we have
a * A = A =
b * A.III. OVER VIEW FCM CATEGORY (FC)
A. Definition
A FCM Category (FC) consist of the following entities /operations ,
1) A collection of attributes /concepts (ie) Att (FC)
a) A collection of Arrows /Relation
b) (ie) Rel (FC)
c) Each Arrows A has asource ‘a’ and target ‘b’ attribute
d) (ie) A: a
b( A is arrow from a to b)2) If a, b and c are any three attributes then Rel (a,b)
Rel (b,c)
Rel (a,c)this operation is written as composition of Arrows
3) Associatively of (FC): A3 composite to (A1 composite to A2) = (A3 composite to A1) to composite A2
4) identity Morphism (
X):For every attributes (relation )X, there exist a morphism
: X
X is called the identity morphismfor X such that for every morphism A : a
b,we have
a* A = A =
b * AIV. FUNCTORS
A functorFf from a FCM Category X to a FCM Category Y, written Ff : X
Y consist of1) for each attributes /concept
a
in X, an conceptF a
in Y and\2) for each morphism A : a
b in X, a morphismF X
:
F a
F b
3) such that the following two properties hold :
4) identity
5) Morphisms
VI. CONCLUTION
Hence we can map by a function from first category to another category successfully main aim of the functoris connect to these.
VII. ACKNOWLEDGEMENT
The authors would like to thank the arrangement of AMET University for its support and encouragement
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